Meshing gears with each pitchline formed of different non-circular curves and a method of obtaining their pitchline profile geometries

ABSTRACT

A pair of driving and driven gears of one-to-one gear ratio, with each gear having an axis of symmetry, and each symmetrical half of the pitchline of the driving gear being formed by at least two different non-circular curves of known polar equations having a junction point at a selected angle from the gear axis of symmetry, and a method of obtaining the pitchline profile geometries of these gears for their production.

Unite States Patent 11 1 1.111 3,721,131

Ingham 1March 20, 1973 MESHING GEARS WITH EACH [56] References Cited PHTCHLHNE FORMED or DIFFERENT UNITED STATES PATENTS NON'URCULAR CURVES AND A 2 957 363 10/1960 1 ham et al 74/393 x ng METHOD OF OBTAINING THEIR 3,125,892 3/1964 Schwesinger ..74/393 PITCHLINE PROFILE GEOMETRIES 3,585,874 6/1971 lngham ....74/393 Field of Search ..74/393, 434

10/1971 Arakawa 74/393 Primary Examiner-Leonard H. Gerin Attorney-Walter Spruegel [5 7] ABSTRACT geometries of these gears for their production.

30 Claims, 10 Drawing Figures PATENTEDMARZO m3 3.721.- 131 sum 10F s PATENTED HARZO I973 SHEET 2 [IF 5 INVENTOR James" 0 figg/za/r/ ATTO N pmimgnmzo ms 3.721.131 sum 3 OF 5 INVENTOR PATENTEDmRzo ms SHEET u nr 5 INVENTOR jamas'fl //7 ATTO E PATENTEDMARZO ms 3. 721, 131 SHEET 5 nr 5 INVENTOR jams'fi 1/29/20? m ATTO NE MESI-IING GEARS WITH EACH PITCIILINE FORMED OF DIFFERENT NON-CIRCULAR CURVES AND A METHOD OF OBTAINING THEIR PITCHLINE PROFILE GEOMETRIES This invention relates to gear drives in general, and to gear drives of cyclically varying angular velocity output in particular.

The present invention is concerned with the type of gear drives involving a pair of meshing driving and driven gears, of which each gear has an axis of symmetry, and the angular velocity of the driven gear varies cyclically according to a given pattern at uniform speed of the driving gear. Various ones of such gears are known, and all of them, except eccentric gears, are non-round gears, with symmetrical halves of the pitchlines being formed by different curves of known polar equations which will impart their inherent varying angular velocity patterns to the driven gears, as in the case of one type or another of an ellipse, for example. However, while these prior non-round gears are entirely satisfactory for many practical drive applications, they are of no avail for a variety of other drive applications involving varying angular velocities of the driven gears. Thus, the velocity ratios attainable with these prior gears are within relatively narrow ranges, for any attempted higher velocity ratio for any particular gear pair would result in the formation in at least one gear of a cusp which would make it most difficult, if not impossible, to form satisfactory teeth thereat. Also, each of these prior gear pairs is limited to one single varying angular velocity pattern which is dictated by the chosen curve of each symmetrical half of the pitchline of the driving gear, wherefore these prior gears are of no avail for gear requirements of two or more different varying angular velocity patterns in a single pair of gears. Further, these prior gears are in their pitchline profiles limitedto known closed curves and, hence, are unavailablefor gears of varying angular velocity patterns afforded by known open curves, such as, for example, a spiral which provides for uniformly varying angular velocity, or constant acceleration and deceleration, of the driven gear.

It is among the objects of the present invention to provide gear pairs of this type which meet any one or more gear requirements that cannot be met by the prior gears in the aforementioned respects of higher velocity ratios, or two or more different varying angular velocity patterns in a single pair of gears, or varying angular velocity patterns afforded by pitchline profiles of known open curves, thereby to make available for the first time gears for many different drive applications which were hitherto unattainable.

It is another object of the present invention to provide gear pairs of this type of which each symmetrical half of the pitchline of the driving gear is formed of at least two different chosen first curves of known polar equations which meet at a junction point at any chosen angle from the gear axis of symmetry, and each symmetrical half of the pitchline of'the driven gear is formed of second curves which are coordinate with the first pitchline curves of the driving gear and also meet at a junction point, with the perimeter of the first pitchline curves being equal to the perimeter of the coordinate second pitchline curves, the first pitchline curves and coordinate second pitchline curves having at their respective junction points the same angles of obliquity, and the angular velocity of the second pitchline curves being the same at their junction point on the drive of the driven gear by the driving gear, all to the end of achieving gear compatibility. With this gear arrangement, it is feasible to provide a pair of gears of a velocity ratio, for example, hitherto unattainable with prior gears of this type because of inevitable formation of a cusp in one of the gears, by selecting one of the first pitchline curves of the driving gear to establish the required varying angular velocity pattern and the velocity ratio of the gears, and making this curve of an angular extent within which no cusp is formed in either gear and the angular velocity pattern requirement for a particular drive application is met, and sacrificing that part of this curve which would lead to a cusp formation in either gear and substituting therefor another first pitchline curve which will avoid a cusp in either gear and whose inherent varying angular velocity pattern is of no consequence in the drive application. It is also feasible to provide a pair of gears of two or even more different varying angular velocity patterns, by forming each symmetrical half of the pitchline of the driving gear of two or more curves selected to establish the different required varying angular velocity patterns, respectively, over the also selected angular extents of the respective curves, with one of these curves also establishing the required velocity ratio of the gears. It is further feasible to provide gear pairs whose required varying angular velocity patterns are attained only with open curves of known polar equations, by forming an intermediate part of each symmetrical half of the pitchline of the driving gear as an open curve to establish the required varying velocity pattern and also the velocity ratio of the gears, with this open curve being of an angular extent within which neither a cusp nor a node is formed in the gears and the angular velocity pattern requirement for a particular drive application is met, and sacrificing those end parts of this open curve which would lead to cusp and node formations in the gears and substituting therefor closed curves which will avoid cusp and node formations in the gears and whose inherent varying angular velocity patterns are of no consequence in the drive application. It is also entirely feasible to provide gear pairs which embody combinations of the described features of the abovementioned feasible gears.

It is a further object of the present invention to provide, for a gear pair of this type which constitute utility gears, a pair of driving and driven balance gears of oneto-one gear ratio, of which driving balance gear turns with the driving utility gear, and the known and selected particular curve part of each symmetrical half of the pitchline of the driving balance gear which is angularly coextensive and, hence, associated with the particular curve part of each symmetrical half of the pitchline of the driving utility gear that establishes the required velocity ratio of the utility gears, is such that during mesh of the gears over these particular pitchline curve parts, and when the mass polar moments of inertia of the driven gears bear a certain known relation to each other, the driven gears are, at constant velocity of the driving gears, in absolute balance at which the rate of interchange of kinetic energy between the driven gears via the driving gears is the same, whereby also the sum of the kinetic energies of the driven gears is at any instant constant. The remaining profile of each symmetrical half of the pitchline of the driving balance gear is formed by a curve or curves selected to achieve in the remainder of the driven gears the best obtainable balance short of absolute balance. With this arrangement, the driven gears are in absolute balance during two periods of each revolution, and are in more or less near balance during the remainder of each revolution, which makes for an overall balance of these gears at which operational torque feed-back from the driven gears into the driving gears and their drive shaft is, at even modern high-speed drive requirements, very low and easily tolerable.

Another object of the present invention is to devise a method of obtaining all necessary particulars of the aforementioned utility gears of this type for their actual production, with the method involving setting up pairs of basic equations, of which the equations of each pair express the lengths of the radii of one of the selected pitchline curves of the driving gear at angles 6, from the gear axis of symmetry and the lengths of their coordinate pitchline radii of the driven gear, with two different constants of unknown values being included in each pair of equations to arrive at equal pitchline perimeters of the gears and at a chosen velocity ratio of the gears to be established over the angular expanse of a single selected one of the pitchline curves of the driving gear, and the constants in the equation pairs being curves of the driven gear, which are coordinate with the selected pitchline curves of the driving gear, at their junction point; setting up still another equation of the angular extent of these coordinate pitchline curves of the driven gear to total 180; expressing in still another equation the chosen velocity ratio by the radius relationship of the selected curve of the pitchline of the driving gear establishing this ratio and its coordinate pitchline curve of the driven gear; from these other equations, and by setting the equations for the angles of obliquity and for the angular velocities at curve junction points equal to each other to satisfy the requirement that they are equal thereat, obtaining the numerical values of all constants in the basic equations; and by using the numerical values of these constants in the basic equations, obtaining actual lengths of the various radii for different angles 0,. In order to obtain for a pair of utility gears of this type a pair of the aforementioned balance gears, the same method is used to obtain the profile geometries of the balance gears.

It is a further object of the present invention to provide utility gear pairs which have the blended curve characteristics of the aforementioned utility gears, but the angular velocity pattern or cycle of the driven gears occurs a plurality of times per revolution of the latter gears for each revolution of the driving gears, or which have a gear ratio other than 1:1, and to apply the aforementioned method for obtaining the profile geometries ofthese gears.

Further objects and advantages will appear to those skilled in the art from the following, considered in conjunction with the accompanying drawings.

In the accompanying drawings, in which certain modes of carrying out the present invention are shown for illustrative purposes:

FIG. 1 is a diagrammatic lay-out of a set of gears of arbitrary pitchline profiles for reference purposes in the derivation of the correct profile geometries of gears embodying the invention;

FIG. 1A is an arbitrary pitchline profile of a gear having intolerable node and cusp formations at its axis of symmetry;

FIG. 2 illustrates the development of correct pitchlines of gears derived with reference to FIG. 1;

FIG. 3 shows the developed gears of the pitchlines of FIG. 2 and their mounting and meshing relationship;

FIG. 3A shows the same gears as in FIG. 3, but in different angular relative positions;

FIG. 4 is a graph depicting featured kinetic-energy characteristics of certain of the gears of FIGS. 3 and 3A;

FIG. 5 illustrates the development of correct pitchlines of a different pair of gears embodying the invention;

FIG. 6 shows the developed gears with the pitchlines of FIG. 5 and their mounting and meshing relationship;

FIG. 7 shows-the pitchlines of another pair of gears embodying the invention in a modified manner; and

FIG. 8 shows the pitchlines of another pair of gears embodying the invention in a further modified manner.

Referring to the drawings, and more particularly to FIG. 1 thereof, the reference numerals 10 and 12 designate arbitrarily drawn pitchlines of a pair of driving and driven gears, respectively, which are to meet certain requirements, and whose profile geometries are to be obtained to permit ready layout of their pitchlines for production of these gears. Thus, the gears must be of one-to-one gear ratio, and each gear must have an axis of symmetry x. It is another, most important, requirement of these gears that each symmetrical half of the pitchline 10 of the driving gear is formed of a number of pairs of successive non-circular curves of known polar equations, of which the curves of each pair have different polar equations, with the number of pairs of successive curves being 2 in this instance, namely the curves c, and c, and the curves 0, and 0,". The pitchline curves c,, c, and 0," have radii r,, r, and r," which are spaced at angles 0, from the axis of symmetry x of the pitchline 10 at which 0, is zero. It is a further requirement of these gears that the pitchline curves c,, c, and 0,, e," have junction points j andj, at given angles 0, equal to A, and M, respectively. Each symmetrical half of the pitchline 12 of the driven gear is formed by curves e c, and c," which are coordinate with the pitchline curves 0,, 4 I and c,", respectively, of the driving gear, and have radii r,, r, and r respectively, at angles 0,, from the axis of symmetry x of the driven gear at which 0, is zero, with the coordinate pitchline curves c, 0,, c, c and c," c," being over their respective angular extents in running mesh with each other once during each revolution of the gears.

In this example, the pitchline curve c, is selected to be part of an open curve which, if extended from 6, (zero) to 0, would leave the pitchline 10, and with it the driving gear, with a cusp and a node, on the gear axis of symmetry x. More particularly, the open pitchline curve c selected in this instance is part of a spiral which, if extended from 6, (zero) to 6, 180 would leave in the pitchline of the driving gear a cusp 14 and a node 16 (FIG. 1A) over the extent of which it would be difficult, if not impossible, to form satisfactory teeth. It is for the purpose of avoiding such cusp and node formations in the driving gear and also in the driven gear, that the spiral curve c,-is an intermediate curve of the pitchline 10, and not an end curve extending either from 0, (zero) or to 19 (180), with the end curves 0, and a," serving in this instance to fill the gaps in the pitchline between 0, (zero) and k, and between A, and 0, (180), respectively. It is, of course, imperative that these end curves 0, and c," are among any known closed curves, for only closed curves can at all avoid cusp or node formations in the pitchline at the gear axis of symmetry.

A spiral has been selected for the pitchline curve c because in this instance it is a primary functional requirement of the gears that the driven gear shall, on rotation of the driving gear at uniform angular velocity, have uniformly varying angular velocity over the extent of its running mesh with the pitchline curve 0 of the driving gear, for this may be achieved only with a spiral curve 0 and its coordinate spiral curve 0 The spiral curve c has thus been selected as the utility" curve which imparts to the gears their intended primary func tion of uniformly varying angular velocity of the driven gear over a selected part of each half-revolution thereof at constant speed of the driving gear, and more 1 particularly constant acceleration and constant deceleration of the driven gear over like parts of successive half-revolutions thereof at constant speed of the driving gear. There-is also the requirement that the gears have a selected velocity ratio V, for their aforementioned primary function, and to this end the spiral curve 0 and its coordinate curve 0 are to have the selected velocity ratio V There is this further requirement of the gears that they have a selected center-tocenter distance g. Finally, the intended primary function of the gears requires uniformly varying angular velocity of the driven gear over a given part of each half-revolution of the gears, and the angles A, and A are selected so that the angular extent of the spiral curve c from A, to A is sufficiently large to meet this requirement. Accordingly, the requirements of uniformly varying angular velocity of the driven gear during each half-revolution of the gears and over the given velocity ratio, are met by selecting the angular extent of the intermediate spiral curve c, and developing this curve and its coordinate curve c so as to have the given velocity ratio, wherefore it is immaterial, insofar as the intended operating function of the gears is concerned, just what varying angular velocity pattern and also velocity ratio the end curves c, and c," and their coordinate curves c and c," establish in the gears. Therefore, the end curves 0, and c of each symmetrical half of the pitchline 10 of the driving gear may be selected from among any closed curves of known polar equations and without any regard to the intended operating function of the gears.

The next objective is to develop general 'polar pitchline profile equations for the respective coordinate pitchline curves c, c c c, and c," 0 Starting with the coordinate pitchline curves 0, c of spiral outline, the relationship of coordinate radii r and r for different anglesfl, of r must vary as a straight line in order to produce the required uniformly varying angular velocity of the driven gear at constant velocity of the driving gear. Thus,

By combining equations (1) and (2), there are obtained the following polar pitchline profile equations for the pitchline curves c and c r =g/(1+a+K0 (4) It will be noted that these equations (3) and (4) are valid only between 0, (zero) and 6 (180). However, since the zero to 180 gear axis is the gear axis of symmetry, a curve identical with the curve 0 between 0 and 180 extends between 180 and 360.

The velocity ratio V, of the coordinate pitchline curves r: and has in this instance been selected to be 2.1:1. It has been found that this velocity ratio can be obtained if the pitchline curve c extends between A, 18 and k without entailing a cusp or node formation in this curve at either end thereof.

Dealing next with the coordinate pitchline curves 0, and c the same have, for convenience, been selected to correspond to the particular pitchline curves of a driving and driven gear in the Hallden US. Pat. No.

2,933,940, dated Apr. 26, 1960. This patent discloses a 7 r,/r =Kcos0+ 1. but this relation, fully expressed, is

1/ z=[ /(B )l( where K is a constant to arrive at a chosen velocity ratio, and B is a constant to arrive at equal pitchline perimeters of the respective gears, with B being equal to 2 in this special case, wherefore l/(B 1) drops out in (6) above and leaves (5) above. However, the constant [3 for r,/r applied to the torque-corrective pitchline curves c, and 0 which must join with the other pitchline curves, namely c and 0 cannot have the value 2 but musthave another value, wherefore the relationship of r 'lr for the pitchline curves c, and 0 must conform to (6) above, and is expressed as n/ z=1 /(B'- 1) (7) where B and K are constants of unknown, but determinable, values to arrive at equal pitchline perimeters of the gears and at the given velocity ratio. The different expressions K cos 6 +1 and 1 Kcos in (6) and (7) above are due to the fact that the gears to which they pertain are 180 out of phase with each other.

With the combined length of coordinate pitchline radii r, and r being equal to the given center-tocenter distance g of the gears, it follows that By combining expressions (7) and (8) above, there are obtained the following polar pitchline profile equations for the pitchline curves c and c t'=g(B )/(B' 1)- The pitchline curve c of the driving gear and its coordinate pitchline curve c of the driven gear may, as already mentioned, be any one among different closed curves of known polar equations, but has in this instance been selected, for convenience and brevity of description, to be the same torque-corrective curve as the coordinate pitchline curves c. and c Accordingly, the polar pitchline profile equations for the pitchline curves c and 0 are similar as the equations (9) and (10), and are as follows,

rlII g(l KII cos 0l)/(BII KHcOS 61), d

Condition A The angular velocity of successive pitchline curves of the driven gear must be the same at their junction point.

The angular velocity of the pitchline curve c of the driven gear can be expressed as where w, is the angular velocity of the driving gear. By substituting for r and r the expressions (3) and (4) therefor, one obtains which reduces to w2=(a+K0 (U1. The angular velocity of the pitchline curve 0 of the driven gear can be expressed as By substituting for r, and r the expressions (9) and (10) therefor, one obtains which reduces to z'=[ /(/3 1) 1 (14) The angular velocity of the pitchline curve 0 of the driven gear can be expressed as By substituting for r," and r the expressions l 1) and (12) therefor, one obtains g(1K cos 6 BK cos 0 which reduces to z"=[ /(B" t) t' (15) The condition that the angular velocity of the driven gear at the junction point of successive pitchline curves of each pair must be the same, can be expressed algebraically by using the equations (13) to (I5) and appropriately substituting A, and A for 0,. Thus (a+K)t )w =[l/(B"1)](l-K" cos M) an (17) where A, 18 and A 165, as selected.

Condition B It is also necessary that the angle of obliquity of successive pitchline curves of the driving gear be the same at their junction point.

The angle of obliquity is generally expressed as From this general expression, the expression for the angle of obliquity d) of the pitchline curve 0, at any angle 0 is developed as follows. Thus, equation (9) expresses r =g (1 K cos OJ/(B K cos 0,)

This expression differentiated, yields the following:

Substituting expressions (9) and (18) in equation 17A), the following is obtained,

d,,' E(T gK sin 0 (6l) BK cos 0,

which reduces to K sin 0,(B1)

tan (1), =[K'sin 0,(B' 1)]/(B'K'cos 0,) (1 -K' cos 0,) In similar fashion, the expressions for the angles of obliquity 0, and b," of the pitchline curves c and 0," respectively, at any angle have been developed, with these expressions being tan 4), =K/(a a+ 2a K0 K0 K 0 (20) and tan (1)," [K" sin 0 (B" l)]/(B" K" cos 0,)

(1K"cos0,). 21 Therefore, by using equations (19) through (21) and appropriately substituting )t, and A for 0,, condition B can be expressed algebraically as follows.

Condition C It is also necessary that the pitchline perimeters of the gears are equal. There will, therefore, be developed an equation which algebraically expresses this condition. Beginning with the coordinate pitchline curves c, and c of the driving and driven gears, their respective lengths must be equal to each other. Thus,

r 'd0 =r (19 I d0 (r 'lr By substituting in the latter equation for r 'lr the expression (7) therefor, one obtains d0 1/(B' 1)](1- K'cos 0, (10,, which in integral form is as follows,

B (24 Thus, equation (24) expresses the angle 0 in radians, of the pitchline curve 0 of the driven gear which is of the same length as the coordinate pitchline curve c, of the driving gear.

in similar manner, there have been developed the following equations,

which expresses the angle 0,, in radians, of the pitchline curve c of the driven gear which is of the same length as the coordinate pitchline curve 0, of the driving gear, and

which expresses the angle 0,, in radians, of the pitchline curve 0 of the driven gear which is of the same length as the coordinate pitchline curve 0," of the driving gear.

By using the equations (24) through (26), condition C may be algebraically expressed as follows,

It will be noted that in the five equations (16), (17),

a (22), (23) and (27-) there are six constants of unknown values, namely a, K, K, K", B and B". However, with the velocity ratio V, of the gears also given, the following relationship is available,

By substituting in the latter equation for r lr the expression l) therefor, one obtains (a+K).,)/(a+K). )=l/V,=1/2.1 where A, and )1 are 18 and 165, respectively. By substituting in equation (28) for A, and A 18 (in radians) and 165 (in radians), the following relationships are obtained,

a 2.01823499 K, and

By simultaneously solving the equations (16), (17), (22), (23), (27) and (29), the following values of the six constants are obtained,

' and of their respective coordinate radii r r, and r,",

for any angle 0, of r,, r, and r may be obtained.

Listed in the following table 1 are actual lengths of coordinate radii r, and r, given at the also listed angles 0 of r from 0 to 180, and calculated on the basis of a selected center-to-center distance g 10 inches, with the listed radii r, and r from 0 to 18 being really the radii r, and r, of the coordinate pitchline curves 0, and c and the listed radii r and r from to being really the radii r," and r of the coordinate pitchline curves c," and 0 Also listed in this table 1 are the actual angles of the listed pitchline radii r These angles 0 were obtained as follows. Thus, equation (24) yields actual angles 0 of radii r which are coordinate with radiir, whose angles 0, are used in the equation, and further yields that angle 0 designated 0 (M), of that radius r which is coordinate with the radius r, of the angle 0, The actual angles 0 of the radii r, of the pitchline curve 0 of the driven gear are 0 ()t,) plus the angles,

obtained from equation (25) between the limits of integration from )t, to 1, of radii r which are coordinate with the radii r, whose angles 0, are used in the latter equation. By adding to the angle 0 ()t,) that angle 0,, obtained from this equation, of the radius r, which is coordinate with the radius r, whose angle A, is used in the equation, one obtains the actual angle 0,, designated 0, (A of the radius r, which is coordinate with the radius r, whose angle is M. Finally, the actual angles 0 of the radii r, of the pitchline curve c," of the driven gear are 0 (A,) plus the angles, obtained from equation 26) between the limits of integration from A, to 1r, of radii r which are coordinate with radii r, whose angles 0, are used in the latter equation.

Further listed in table 1 are values of (r,/r for the listed angles 0,, with these values serving for a purpose explained later.

The listed radii r, and r, are for the present purpose sufficient in number for fairly accurate layout in FIG. 2 of the actual profile of one symmetrical half of each of the pitchlines l0 and 12 of the driving and driven gears at a reduced scale, with the radii r, and r being there laid out at their listed angles 0, and 0 Further, with each pitchline and 12 having an axis of symmetry at which the respective angles 0, and 0, are zero and 180, the listed values of r, and r, and their listed angles 0 and 0, may be used for equally accurate layout of the other symmetrical halves of the pitchlines l0 and 12. The actual driving and driven gears 18 and 20 are illustrated in FIG. 3 and 3A in different relative angular positions, and their respective pitchlines l0 and 12 are there shown in dot-and-dash lines.

The present invention lends itself broadly to the determination of the pitchline profile geometries of any pair of driving and driven gears of one-to-one gear ratio, of which each gear has an axis of symmetry, and each symmetrical half of the pitchline of the driving gear is formed by two or more successive non-circular curves of known polar equations, of which successive curves have different polarequations, and the curves may include an open curve or curves which, however, must be arranged intermediate closed end curves to avoid cusp and node formations in either or both gears. To demonstarte this further, there are to be determined the profile geometries of another pair of driving and driven gears whose respective pitchlines 22 and 24 are shown in FIG. 5. Thus, each symmetrical half of the pitchline 22 of the driving gear is to be formed by two successive curves c, and c, of radii r, and r,. The pitchline curves c, and 0,, being end curves, must be closed curves of known polar equation, and being also successive curves, must have different polar equations, and these curves have a junction pointj at the angle 0 equal to A, which is selected to be from the gear axis of symmetry at which 0, is zero and Each symmetrical half of the pitchline 24 of the driven gear is to be formed by curves c, and c, which are coordinate with the pitchline curves c, and 0,, respectively, and have radii r, and r spaced at angles 0 from the gear axis of symmetry at which 0, is zero and 180.

For the exemplary desired utility function of these gears, it is required that, at uniform angular velocity of the driving gear, the driven gear has over each halfrevolution varying angular velocities of two different patterns established by the selected pitchline curves 0, and c,, respectively, of the driving gear. Thus, the one pattern of varying angular velocity of the driven gear established by the pitchline curve 0, is expressed by m =w, Va+K cos 0,, (30) and the other pattern of varying angular velocity of the driven gear established by the pitchline curve c, is expressed by w,=w,(a'+K'cos0,), (31) where a,a'., K and K are constants of unknown values to arrive at equal pitchline perimeters, and at a given velocity ratio, of the gears.

The varying angular velocity pattern expressed by equationv (30) is the same as that established by symmetrical halves of the pitchline profiles or curves of a driving gear and one driven gear which are shown, and whose developed profile geometries are described, in my prior U.S. Pat. No. 2,957,363, dated Oct. 25, 1960. The other varying angular velocity pattern expressed by equation (31) is established by using the aforementioned torque-corrective curve as the pitchline curve 0, of the driving gear.

The selected velocity ratio of the gears is V,- 2:1, and this velocity ratio is to be from 0, (zero) to 0, (k,); i.e., over the extent of the pitchline curve 0, of the driving gear. Finally the center-to-center distance g of the gears is to be 10 inches.

Following the earlier procedure in developing the profile geometries of the gears 18 and 20 with their pitchlines l0 and 12, the polar pitchline profile equations for the coordinate pitchline curves c, c, and c, c will be set up.

Thus, and with respect to the coordinate pitchline curves 0, c,, the angular velocity of the driven gear over the extent of its pitchline curve a, is

It follows from expression (30) that from which one obtains /(l Va +K cos 6,).

With respect to the coordinate pitchline curves c,

c the angular velocity of the driven gear over the extent of its pitchline curve c is from which one obtains r =g (a' K cos )/(1 +a' K cos 0,) 34 further (g r ')/r a K cos 0,, from which one obtains r '=g/(1 +a+ K cos 0,). (35) .The condition that the angular velocity of the pitchline curves c and c, at their junction point is equal, is by the equations (30) and (31) algebraically expressed as follows,

ms'T,=a'+K' cos A, (36) The gears must satisfy the further condition that the angle of obliquity of the pitchline curves c and c, of the driving gear at their junction point is the same. The angle of obliquity of the pitchline curve c, at any angle 0 is 4 1 1)'( r1/ 1) By differentiating expression (32) and multiplying it by 1/ r one obtains tan 1=K sin 6,/2(l Va-i-K cos 01) (a-l-Kcos 0) Va K cos 0, 110,, or

Equation (40) is not solvable in this form. Recourse is, therefore, had to the trigonometric identity cos 0, =1 2 sin (0 /2) By making (6 /2) B, and using expression (41 (K/a) cos 0 =(K/a)- (2K/a) sin fl (42) By using expression (42) in equation (40),

a M 0 =2] +K 2 f.

2 0 a+K Bdfi 43 Equation (43) is an elliptic integral of the second kind which, for solution, is expressed as follows:

The relationship between a and K is established once the I following radius relationship (r,/r for the given 5 velocity ratio is established.

The angle of obliquity of the pitc hline curve c, at any angle 6 is same, is by the equations (37) and (38) algebraically expressed as follows,

'K sin )t,/2( 1+ V a+K cosh (a+K cos A =K' sin )\,/(1 +a' +K' cosh (a'+K' cosh 39 The gears have to satisfy the further condition that the pitchline perimeters of the gears are equal. Thus, in connection with the coordinate pitchline curves c, and it is known that where (9 and '0, are coordinate angles of r and r respectively.

Therefore,

(10 (f /r d0 By substituting for (r /r the expression (31a) therefor,

I /a+K cos 0,)0 =0 wa+K cos 0 )0 =130 By squaring this expression and using the cosine values for 0 and 130, one obtains (a K) /(a 0.64278761 K) 4, or

a K 4a 2.57115044K, which yields a= 1.19038348K 4s K= 0.84006542a. (46a) By using these relations ofa and k in expression (45) for h, i

h V(2K)/( 1 .19038348K+ K) V0.9l3082l2,

From Tables of Elliptic Integrals, the value of E for the determined values of B and h is 0 Substituting this value of E in equation (44),

The angular extent of the pitchline curve a, is ex pressed as fi01= fla' K cos 0,) do,

and by integrating this equation,

[8'0 K sin 0,1 (48) The condition that the pitchline perimeters of the gears must be equal is by the equations (47) and (48) algebraically expressed as follows.

To obtain the values of the constants a, a, K and K, equations (36), (39), (46) and (49) are solved as follows.

By substituting in equation (49) a for K as expressed in equation (46), and integrating expression (48), expression (49) becomes 2 (1.26349753) VE+a2-12.2689277Oa' -0.76604444K' 7ror 50) 2 (1.26349753) a O.87266495a.76604444 K 7r.

By substituting a for K (46), and using die expression for Va +K cos it, (36), in equation (39), the latter becomes which reduces to 0.84006542a =2K (a+K cos A It follows from equation (36) that a= Va +K cos M-K cos A and by substituting a for K, and using the value of cos r( Expressions (51 and (52) are next combined as follows:

0.84006542a =2 K ]O.67824505 VE+O.64278761K which reduces to K=0.84006542a 1.35649010 V71 and further reduces to K'=0.6l929343 V71. 53 Substituting this expression for K in equation (52), the latter reduces to a=l .0763 1919 Va. 54 Substituting in equation (50) the expressions (53) and (54) for K and a, there is obtained 2.52699506 VZ+O.87266495(1.07631919) Vii O.6l929343 (76604444) V2T=1r Listed in the following table II are actual lengths of coordinate radii r and r given at the also listed angles 6 for r, from 0 to 180, and calculated on the basis of the selected center-to-center distance g 10 inches, with the listed radii r and r from to being really the radii r, and r of the coordinate pitchline curves 0, and c Also listed in the table are the actual angles 0 of the listed radii r,. The angles 0 of the listed radii r which are coordinate with the listed radii r over the 0 range from 0 to 130 were obtained from equation (44), 0 =2 Va +K E(h,0 /2), where h==0.95555330. The angles 0 of the listed radii r which are coordinate with the listed radii r over the 0 range from 130 to 180, were obtained from the following equation 0 =152.6325OO46 6 a13Oa'+(K sin 6 /O.O1745329) (K sin 130/0.0l 745329).

The listed radii r, and r are for the present purpose sufficient in number for fairly accurate layout in FIG. 5 of the actual profile of one symmetrical half of each of the pitchlines 22 and 24 of the driving and driven gears at a reduced scale, with the radii r and r being there laid out at their listed angles (9 and 0 Further, with each pitchline 22 and 24 having an axis of symmetry x at which the respective angles 0 and 0 are zero and 180, the listed values of r and r and their listed angles 0 and 0 may be used for equally accurate layout of the other symmetrical halves of the pitchlines 22 and 24. The actual driving and driven gears 26 and 28 are illustrated in FIG. 6, and their respective pitchlines 22 and 24 are there shown in dot-and-dash lines.

The described exemplary gear pairs 18, 20 and 26, 28 embody two important aspects of the present invention. One of these aspects of the invention lies in the provision of meshing gear pairs of one-to-one gear ratio, of which each gear has an axis of symmetry, and each gear pair has the unique characteristic that each symmetrical half of the pitchline of one gear thereof is formed by a number of pairs of successive non-circular curves of any known polar equations, of which the curves of each pair have different polar equations and a junction point whose pitchline radius forms a given angle with the gear axis of symmetry. Thus, in the described gear pair 18,20 the number of pairs of successive non-circular curves of each symmetrical half of the pitchline 10 of the driving gear 18, is 2, all of these curves have known polar equations, and the successive curves of each pair are different from each other and have different polar equations. Moreover, this gear pair demonstrates the further unique feature of permissive selectability of a known open pitchline of the driving gear, with this open curve bearing, however, arranged intermediate end curves which must be closed curves to avoid at all cusp and node formations in the gears. In the other described gear pair 26, 28 the number of pairs of successive non-circular curves of each symmetrical half of the pitchline curve 22 of the driving gear 26, is 1, these curves have known polar equations, and the successive curves of the one and only pair are different from each other and have different polar equations.

The aforementioned other aspect of the invention lies in a method according to which the pitchline profile geometries of the exemplary gear pairs 18-20 and 26-28, and many other gear pairs, are obtained. To use this method for obtaining the pitchline profile geometries of a pair of gears, the following particulars of the gears must first be given:

1. the center-to-center distance of the gears,

2. one symmetrical half of the pitchline of the driving gear is to be formed by a given number of pairs of successive selected non-circular first curves of known polar equations, of which the curves of each pair have different polar equations and a junction point at a selected angle 0, from the gear axis of symmetry at which is zero, and the driven gear is to have second pitchline curves coordinate with these first pitchline curves;

3. a selected one of these first pitchline curves and its coordinate second curve of one symmetrical half of the pitchline of the driven gear, are to have a selected velocity ratio.

With these particulars of the gears given, one may proceed with the following featured steps of the method in the order in which they are here recited or in any other feasible order: a setting up first pairs of equations, of which the equations of each pair express the lengths of the radii of one of the aforementioned first pitchline curves at angles 6 and the lengths of their coordinate pitchline radii of the driven gear, and include two constants of unknown values to arrive at equal pitchline perimeters of the gears and at the selected velocity ratio, with the constants in the equation pairs being also different from each other;

setting up a pair of second equations for the angular velocities of successive second pitchline curves of each pair at their junction point;

setting up a pair of third equations for the angles of obliquity of successive first pitchline curves of each pair at their junction point;

setting up a fourth equation of the angular extent of the aforementioned second pitchline curves to total 180;

expressing in a fifth equation the selected velocity ratio by the radius relationship of the aforementioned one first pitchline curve and its coordinate second pitchline curve;

from the aforementioned second through fifth equations, and by setting the second and third equations of each pair equal to each other, obtaining the numerical values of all constants;

and by using the numerical values of the constants in the aforementioned first pairs of equations, obtaining the actual lengths of coordinate pitchline radii of the gears for different angles 6 This method is general for the determination of the pitchline profile geometries of any gear pair of which one symmetrical half of the pitchline of the driving gear is formed by two or more curves, of which successive curves of each pair we have known, but different, polar equations, and a velocity ratio of the gears is to be established by a selected one of these curves and its coordinate pitchline curve of the driven gear. Thus, it follows from the description herein of the exemplary gear pairs 18, 20 and 26,28 that the method did indeed lend itself to the determination of their pitchline profile geometries, whereby in the case of the gear pair 18,20 the given number of pairs of successive first curves of one symmetrical half of the pitchline of the driving gear, called for by the method, was 2, whereas in the case of the gear pair 2 6, 28 this number was 1. The method is also general for the determination of the pitchline profile geometries of any gear pair of which a plurality of curves, which form one symmetrical half of the pitchline of the driving gear, includes one or more opencurves, such as a spiral curve, for example, except that for the opposite end curves of this half-pitchline there must be selected the same or different known closed curves, such as the same or different elliptic curves, or the described exemplary torque-corrective pitchline curves of the gear pair 18 and 20, for example, in order to avoid at all cusp and node formations in the gears.

By the present method, there may be obtained the pitchline profile geometries of gear pairs for many different utility purposes. Thus, the gearpair 18, 20 affords constant acceleration and constant deceleration of the driven gear over required parts of successive half-revolutions thereof at constant angular velocity of the driving gear, which is an achievement hitherto unattainable with any gear pair. The other described gear pair 22, 24 is but an example of gear pairs of which the driven gears have for each revolution thereof at least two different required patterns of varying angular velocities, which is another achievement hitherto unattainable with any gear pair. There is also the case where the pitchlines of a pair of gears are to be formed by a single known closed curve, such as the aforementioned torque-corrective curve, for example, but a selected velocity ratio would result in cusp formations in the gears. in that event, the cusp-forming parts of the pitchlines can be sacrificed and in their place used different known blending curves which avoid cusp formations, with the present method serving to obtain the pitchline profile geometries of the gears.

ltwill be noted that with the present method as expressed above there are obtained the lengths of coordinate pitchline radii r, and r: of the driving and driven gears for any angle 0 of r,. Therefore, taking into consideration that the angles 0 of pitchline radii r, of the driven gear are different than the angles 0 of their coordinate pitchline radii r, of the driving gear, it stands to reason that only by laying out many pitchline radii r of the driven gear in close peripherally spaced relation with each other and with the coordinate pitchline radii r, of the driving gear will the pitchline of the driven gear be outlined accurately. However, the

task oflaying out the pitchline of the driven gear will be greatly facilitated by determining the angle of each determined pitchline radius r of the driven gear. To this end, the present method further comprises setting up expressions for the angles 6 2 of the radii of the aforementioned second pitchline curves, respectively, of the driven gear in terms of the angles (9 of their coordinate pitchline radii of the driving gear, from which to obtain the actual angles 0 The present method also lends itself to the determination of the pitchline profile geometries of a pair of balance gears for a pair of utility gears of the described type of which one symmetrical half of the pitchline of the driving gear is formed by at least two different known curves. It is, therefore, an exemplary objective to obtain a pair of balance gears for the described utility gears 18 and 20 of FIGS. 3 and 3A whose pitchlines 10 and 12 are laid out in FIG. 2 and also arbitrarily shown in FIG. 1. Referring to FIG. 1, there are shown in arbitrary outline the pitchlines 30 and 32 of the driving and driven balance gears, of which the driving balance gear is mounted on the same shaft 34 on which the driving utility gear is mounted. It must be stated at the very outset that, due to the different curves of each symmetrical half of the pitchlines 10 and 12 of the utility gears, it is impossible to provide a pair of balance gears which achieve absolute dynamic balance between the utility and balance gears at any instant of their drive. However, it is possible to provide a pair of balance gears which are in part designed so that they will, during running mesh of a selected one only of the different curves of each symmetrical pitchline half of the driving utility gear with its coordinate pitchline curve of the driven utility gear, achieve absolute balance between the gears at which the rate of interchange of kinetic energy between the driven gears via the driving gears is the same, whereby also the sum of the kinetic energies of the driven utility and balance gears is constant at any instant on rotation of the driving gears at constant angular velocity. As pointed out more fully hereinafter, the remaining part of the balance gears may be designed so that they will achieve more or less near balance of the gears during running mesh of the other curves of each symmetrical pitchline half of the driving utility gear with their coordinate pitchline curves of the driven utility gear.

Because the curve c, of each symmetrical pitchline half of the driving utility gear is of greater, and in this instance of far greater, angular extent than either of the remaining curves 0, and c,", it is, of course, obvious to select this pitchline curve c as the one over whose running mesh with its coordinate pitchline curve c, of the driven utility gear the balance gears are to establish absolute dynamic balance between thegears. In my copending application Ser. No. 852,363, filed Aug. 22,

1969, there are developed general equations for the pitchline profile geometries of a pair of balance gears for a pair of utility gears of a variable angular velocity pattern of the driven gear inherent in the selection of a particular curve of known or determinable polar equation for the pitchlines of the utility gears. Thus, in this copending application, the radii of each symmetrical half of the pitchlines of the driving and driven utility gears are designated r and r respectively, and the radii of each symmetrical half of the pitchlines of the driving and driven balance gears are designated as r and r respectively, and it was found that absolute balance between the gears is achieved if the radius relationship of the gears satisfies the basic equation 1/ 2)+ 3/n)= (55) where r,, r and r r are coordinate radii at the same angles 0 and 0 for r, and r respectively, and x and Z are constants of unknown, but determinable, values.

This equation (55) reflects the relation of the magnitudes of the kinetic energies of the driven utility and driven balance gears at any constant velocity of the common drive shaft and the gears turning therewith, and expresses that the sum of the kinetic energy of the driven utility gear plus the kinetic energy of the driven balance gear must be equal to a constant at any instant. Insofar as the gear system of FIG. 1 in the present application is concerned, equation (55) can apply only over the extent of simultaneous mesh of the curve c, of each symmetrical pitchline half of the driving utility gear and of its associated curve 0 of each symmetrical pitchline half of the driving balance gear with their respective coordinate pitchline curves c and c of the driven utility and balance gears. Accordingly, the pitchline curve a of the driving balance gear, which is associated with the pitchline curve 0 of the driving utility gear, is in FIG. 1 shown as extending between the same angles )1, and A between which the pitchline curve 0 of the driving utility gear extends, with the pitchline curve c having radii r;,, and its coordinate pitchline curve 0 of the driven balance gear having radii r By substituting expression (1) for r /r in the basic equation (55), the same becomes (a 1(0) x (r /r Z, from which follows that With the center-to-center distancebetween the driving gears and driven gears being g 10 inches as selected,

By using the latter expressions for r;, and r, in equation (56), one obtains 3 96+ z-(u+1 a. 57 and x+ /Z (a-l-Kl9 (5 These equations (57) and (58) are the pitchline profile equations of the driving and driven balance gears between the angles 0 and 0 M, and these equations have been set up pursuant to the earlier featured method.

Next, and in further pursuance of the featured method, the pitchline profile equations of the coor-' dinate blending curves c 0, and c 0 of the driving and driven balance gears are determined, with the blending curve c being associated with the pitchline curve 0, of the driving utility gear and extending over the angle 6, (M) which is equal to the angle 19 (A of the pitchline curve c andthe blending curve c being associated with the pitchline curve 0," of the driving utility gear and extending over the same angle as the latter, this being between A, and 6, for curve 0," and being between A, and 0:, (180) for curve c,. The coordinate pitchline curves c and c of the balance gears have radii r and r respectively, and the coordinate pitchline curves and c of the balance gears have radii r "and r respectively. The curves selected for c and c,," are preferably and advantageously the same as their associated pitchline curves c and c of the driving utility gear, with these curves being the aforementioned torque-corrective curves. Accordingly, the pitchline profile equations to be set up for the coordinate pitchline curves c c and c c of the balance gears are the same as those for their respective associated pitchline curves c c [equations (9) and (10)] and associated pitchline curves 0,, c [equations (11) and (12)], except that the determined values of the constants K, K", B and [3 in equations (9) through (12) for the utility gears are obviously different from those to be determined for the balance gears, wherefore the constants K, K", [3 and B" in the profile equations to be set up are carried as k, k", a and a", respectively, and some of the plus and minus signs in the equations (9) to (12) will also have to be reversed in the equations to be set up because the balance gears are 180 out of phase with the utility gears. Accordingly, the pitchline profile equations of the blending curves c and c, are

r, g (al )/(a+kcos 0 (60) and the pitchline profile equations of the blending curves 0;," and c, are

In further pursuance of the featured method, there will be set up expressions which algebraically express the following conditions:

1. the angular velocity of the pitchline curves c and c of the driven balance gear must be the same as their junction point; 1

2. the angular velocity of the pitchline curves c, and c of the driven balance bear must be the same at theirju nction point; i

3. the angles of obliquity of the pitchline curves c and c of the driving balance gear must be the same at theirjunction point;

4. the angles of obliquity of the pitchline curves c and c of the driving balance gear must be the same at their junction point; and

5. the pitchline perimeters of the balance gears must be equal.

' As to condition l above, it is known that the angular velocity, of the driven balance gear over theextent of the pitchline curve Q, is

With the driving balance gear being on the same shaft 34 as the driving utility gear, (0 is equal to :0 and by making (0 6111181120 1,

Similarly, the angular velocity of the driven balance gear over the extent of the pitchline curves, is

w =r /r.,'.

Since the angular velocity of these two pitchline curves must be the same at their junction point, this is expressed as By using the expression (56) for r /r and using for r 'lr the expression (7), but'substituting a and k for B and K and changing plus to minus in the bracket to make the equation applicable to the balance gears, equation (63) becomes To express condition 1 above, A, is substituted for 0,, thus (l/x) \/Z(a+K)t =[1/(a-1)](1+k' cos )t,), 65 Similarly, condition 2 above is expressed algebraically as follows:

As to condition 3 above, the angle of obliquity of the t pitchline curve c of the driving balance gear at its juncand the angle of obliquity of the pitchline curve 6 of Similarly, condition 4 above is algebraically expressed as follows:

zK(a+Ic)\ [Z-l-N L) ]I (a+K \2) iwww [oz-llc cos h jll Ha cos M] Condition 5 above is algebraically expressed as follows:

There are now five equations (65) through (69) with eight constants, a, K, x, -Z, k, k"a and a". With the values of a and K being the same as those determined 1 for the utility gears, there is required a sixth expression to obtain the values of the six remaining constants. The required sixth expression is available once a velocity ratio has been selected for the balance gears. As explained in the aforementioned copending application Ser. No. 852,363, the pitchline curve there developed for the balance gears, which has been used for the coordinate pitchline curves c and c of the present balance gears, permits a velocity ratio which may be the same as, ordifferent from, that of the utility gears. Accordingly, the velocity ratio of the balance gears is to be established by and over the extent of the coordinate pitchline curves c and c of the balance gears, because these pitchline curves afford a choice of a velocity ratio at which the balance gears will be cuspless and which may or may not be the same as the velocity ratio of the utility gears. In this case, it has been found that by selecting a velocity ratio of 1.55 for the balance gears, the latter will be cuspless. Using this velocity ratio value, we have the following algebraic relation:

[ii/5W] [gvmml which reduces to NZ- (a+Ko. ]i. 1. 55 1) i. (71) With a=0.56208954, and K=0.2785055, as determined for the utility gears, equation (71 yields Solution of the equations (65) through (69) yields the following values of the remaining constants:

By using the determined values of these constants in the pitchline profile equations (57) through (62), the actual lengths of radii r r r;," and of their coordinate radii r r and r,", for any angle 0 of r r and r may be obtained.

Listed in the following Table III are actual lengths of coordinate pitchline radii r and r given at the also listed angles 0 for r, from 0 to 180, and calculated on the basis of the given center-to-center distance of the balance gears g= inches, with the listed radii r and r from 165 to 180 being really radii r," and n", and the listed radii r and r from 0 to 18 being really radii r and r TABLE 111 Also listed in this table 111 are the calculated angles 0 of the listed radii r For their calculation,'recourse was had to the following expressions which were taken from equation (69).

law an fi 2K VZ (ad-K0 stn {Z M,

and 1 r m [OI-1'70 Sin 61]) The listed angles 0 were obtained from the equations (72) through (74) in the same explained manner in which the listed angles 0, in table I were obtained from equations (24).through (26).

The radii r, and r listed in table [II are for the present purpose sufficient in number for fairly accurate layout in FIG. 2 of the actual profile of one symmetrical half of each of the pitchlines 30 and 32 of the driving and driven balance gears at a reduced scale, with the radii r;, and r being there laid out at their listed angles 0 and 0 Further, with ,each pitchline 30 and 32 having an axis of symmetry x at which the respective angles 0 and '0 are zero and 180, the listed values of r, and r. and their listed angles 0 and 0 may be used for equally accurate layout of the other symmetrical halves of the pitchlines 30 and 32. The actual driving and driven balance gears 36 and 38 are illustrated in FIGS. 3 and 3A, and their respective pitchlines 30 and 32 are there shown in dot-and-dash lines.

Following is a determination of the dynamic balance achieved in the gears. Thus, it has already been explained that absolute balance between the gears is achieved wherever over the extend of their pitchlines their radius relationship satisfies the basic equation Thus, the values of (T1/Tz) and of at(r=,/r.,) reflect the true relation of the magnitudes of the kinetic energies of the driven utility gear 20 and the driven balance gear 38, respectively, when the mass polar moment of inertia I of the driven utility gear and the mass polar moment of inertia I. of the driven balance tive driven utility and balance gears for like angles 61 and 63, and accordingato the dictates of basic equation (55), their sum for equal angles 6, and 6 must be equal to Z.

TABLE [V 01 6:1 (Ti/r21 l-x (r -;/r A( '1/ 2) 41x2 's/ In this table IV, there are listed under the heading the added values of (r /r and x (r /r given in the respective tables I and III for the same angles 0 and 6 and it will be noted in this table IV that their sum from the angles 6,, 0 18 to the angles 6 6 165 is indeed constant, with this constant being exactly 2.88678667, i.e., the calculated value of Z. The sums of these kinetic energies listed in table IV between 18 and 165 deviate slightly from the exact value 2.88678667, but they would exactly total the latter value at still greater accuracy of the listed values of (r,/r and x (r /r in the respective tables I and 111. Accordingly, at constant velocity of the driving gears 18 and 36, the driven gears 20 and 38 are in absolute balance, and the sum of their kinetic energies is constant at any instant, during intermesh of coordinate pitchline curves c c and c c. of the gears.

The sums of the kinetic energies of the driven gears listed in table IV between 6 0 0 and 0 6 18 and between 6 ,0 165 and 6 6 180 deviate notably from the calculated value of Z, as was fully expected because these energies are developed during intermesh of the coordinate pitchline curves c c and c c of the balance gears, and these curves by being different from the pitchline curves a c cannot follow the dictate of the basic equation (55) insofar as the sum of the kinetic energies being constant at any instant is concerned. It is nevertheless noteworthy that for the relatively small angular extents of the blending pitchline curves c and c of the driving balance gear between 0 and 18 and between 165 and 180, the listed sums of the kinetic energies of the driven gears, while notably deviating from the calculated value of Z, in reality are very close to the latter value, meaning that the driven gears are even over intermesh of the blending pitchline end curves of the gears in nearly absolute balance.

Two further columns in table IV, headed by A(r,/r

and Ax (r /r list values which represent changes in the listed representative kinetic energies of the driven utility and balance gears between listed angles 0 0 with these values being obtained by subtracting successive listed values of (r /r and of x (r /r in the respective tables 1 and Ill. It will be noted in table IV that the values of the changes of the kinetic energy of the respective driven gears are for all practical purposes equal and of opposite sign from 0 0 18 to 0 0 165, and these values would be exactly equal at still greater accuracy of the listed values of (r,/r and x (r /r in the respective tables I and 111. As expected, the listed values of the changes of the kinetic energies of the respective driven gears from 0,, 0 0 to 0 6 80 and from 6 0 165 to 6 6 180, while of opposite sign, are notably different from each other, though they are in reality quite close to each other.

The kinetic energies of the driven utility and balance gears are also plotted in FIG. 4 for 180 rotation of the drive shaft 34, with the kinetic energies being of the values (r lr and x (r /r listed in the respective Tables I and III. The sum of these kinetic energies is represented in FIG. 4 by the line L at the level shown with respect to the calculated value 2.88678667 of Z. However, this line has been shifted to a lower level and designated L in order to illustrate its full length. It will be noted that the line L between 18 and 165 indicates that the sum' of the kinetic energies of the driven gears over this angular range is indeed constant and equal to the calculated value of Z, whereas the ends of this line between 0 and 18 and between 165 and 180 indicate by their slight deviation from the long straight part of this line that the sum of the kinetic energies represented by them varies from the calculated value of Z as their corresponding listed sums of kinetic energies in table IV.

The balance gears 36, 38 are but one example of applying the featured method to the determination of the pitchline profile geometries of a pair of balance gears for any pair of utility gears of which each symmetrical half of the pitchline of the driving gear is formed by at least two selected different curves of known or determinable polar equations.

While each of the described utility gears 18, 20 and 26, 28 of FIGS. 3 and'6, respectively, has a single axis of symmetry so that each driven gear undergoes one cycle of varying angular velocities during each revolution of the associated driving gear, it is fully within the purview of the present invention to provide, according to the featured method, a pair of driving and driven utility gears of which the driven gear has a plurality of cycles of varying velocities for each revolution of the driven gear. Thus, FIG. 7 shows the pitchlines 34 and 36 of a pair of driving and driven gears, respectively, of which the driven gear undergoes an exemplary number of two successive cycles of varying angular velocities per each revolution of the driving gear, meaning that one such cycle is to occur during each half-revolution of the gears, wherefore the gears will have two axes of symmetry x and x. Further, let the pitchline of the driving gear be formed of two blending curves c and c of known but different polar equations for each half-cycle of varying angular velocities, with these curves c and c, being in this instance of the described polar equa-, tions of the pitchline curves c, and c of the pitchline 22 of the driving gear 26 of FIGS. 5 and 6. Let it further be assumed that in this example the curve c extends from 19 0 to 6 80, and that the curve c, extend from 0, 80 to 0, 90, and let the center-to-center distance of the gears be g= inches. 7

In accordance with the featured method, the profile geometries of these gears are obtained as follows. The angular velocity of the driven gear is to vary as w m, V a K cos 26, from 0, (0) to 0 (8O), 75

and as m m, (a K cos 26,) from 0 (80) to 6, (90)(76) The polar pitch profile equations of the curve c and of its coordinate curve 0 of the pitchline of the driven gear, are

and 8) Accordingly,

(K/a) cos 2 6, (K/a) (2K/a) sin 6 wherefore equation (84) may be written as:

The polar pitch profile equations of the curve c, and

of its coordinate curve 0 of the pitchline of the driven gear, are

The condition that the angular velocity at the junction of the pitch curves c and 0 is equal is expressed algebraically as follows.

whe re A 80.

The condition that the angle of obliquity of the pitch curves c and c, at their junction pointj is the same is expressed algebraically as follows.

( 2 K')/(l +a +K cos 2A) (a' +K cos 2A K)/(l V aKcos 2A)(a+Kcos 2A) (82) The gears must satisfy the further condition that their d6 la K cos 2 9,116, ,hence 0 I 6 /af COS 201d0 This equation is not solvable in this form, wherefore recourse is had to the trigonometric identity 'cos 2 6,= l 2 sin 0,.

solution is expressed as follows:

The relationship between a and K is established once a velocity ratio has been specified. Let the velocity ratio be V 1.5. Accordingly,

by squaring both sides of this equation, one obtains (a K)/(a0.93969262 K 2.25 wherefore a K= 2.25a 2.11430839 K, or

l.25a= 3.11430839 K therefore a=2.49l4467l K, (88) K=O.40l37322a (89) By using the above relations in the expression (87), one obtains h= \l2K/(249l4467l K+ K) V0.57282844 or h=0.75685430. For 6 80, and h equal to the value just above, the

value of E is obtained from Tables of Elliptic Integrals,

and is as follows:

Therefore, by using this value in equation (86), one obtains 0 Va+K(l.l98l50l6). 90 The angular extent of the pitchline curve c is expressed as:

90 0 =J;\ (a COS 2601101 0,: [a' 6,+(K/2) sin 2 0,133 92) Therefore, for pitchline compatability for 90 rotation of the driving gear, the following must prevail: 

1. Method of obtaining the profile geometries of a pair of driving and driven gears meshable at a given center-to-center distance and each having an axis of symmetry, with one symmetrical half of the pitchline of said driving gear being formed by a number of pairs of successive non-circular first curves of known polar equations, of which the curves of each pair have different polar equations and a junction point at a given angle theta 1 from the gear axis of symmetry at which theta 1 is zero, and said driven gear having second pitchline curves coordinate with said first pitchline curves, and a given one of said first pitchline curves and its coordinate second pitchline curve having a given velocity ratio, said method comprising setting up first pairs of equations, of which the equations of each pair express the lengths of the radii of one of said first pitchline curves at angles theta 1 and the lengths of their coordinate pitchline radii of said driven gear, and include two different constants of unknown values to arrive at equal pitchline perimeters and at the given velocity ratio, with the constants in said equation pairs being also different from each other; setting up a pair of second equations expressing the angular velocities of successive second pitchline curves of each pair at their junction point, with said second equations including the constants in the length expressions of the radii of the respective second pitchline curves; setting up a pair of third equations expressing the angles of obliquity of successive first pitchline curves of each pair at their junction point, with said third equations including the constants in the length expressions of the radii of the respective first pitchline curves; setting up a fourth equation expressing the combined angular extent of said second pitchline curves in terms of theta 1 to total 180* with said fourth equation including the constants in the length expressions of the radii of the respective second pitchline curves; expressing in a fifth equation the given velocity ratio by the radius relationship of said given first pitchline curve and its coordinate second pitchline curve, with said fifth equation including the constants in the length expressions of the radii of thE respective first and second pitchline curves; from said second to fifth equations, and by setting said second and third equations of each pair equal to each other, obtaining the numerical values of all of said constants; and by using the numerical values of said constants in said first pairs of equations, obtaining the actual lengths of coordinate pitchline radii of said gears for different angles theta
 1. 2. The method of claim 1, in which said number of pairs of successive first pitchline curves is
 1. 3. The method of claim 1, in which said number of pairs of successive first pitchline curves is greater than
 1. 4. The method of claim 1, in which said first pitchline curves are closed curves.
 5. The method of claim 3, in which said first pitchline curves are of open and closed types.
 6. The method of claim 5, in which a first pitchline curve of open type and its coordinate second pitchline curve have said given velocity ratio.
 7. The method of claim 1, in which said number of pairs of successive first pitchline curves is 2, and said first pitchline curves are two closed end curves and an intermediate open curve.
 8. The method of claim 7, in which said intermediate pitchline curve is part of a spiral.
 9. The method of claim 1, in which one side of said fourth equation is equivalent to 180*, and the other side of said fourth equation is formed by additive expressions which denote the angular extents of said second pitchline curves, respectively, and are integrated between the limits of theta 1 of their coordinate first pitchline curves, with said method further comprising obtaining angles theta 2 of radii of said second pitchline curves, by obtaining sets of angles in solving said expressions for various angles theta 1 within their respective limits of integration, and adding to each set of angles thus obtained the angle obtained in solving the respective expression for the lower limit of its integration.
 10. Method of claim 1 for obtaining the profile geometries of a pair of driving and driven gears of which said one symmetrical half of the pitchline of said driving gear extends from theta 1 0* to theta 1 180* at the gear axis of symmetry on opposite sides, respectively, of the gear center, and said one symmetrical pitchline half of said driving gear has a whole number n of successive lengths of identical curvature and angular extent, of which the one pitchline length starting at theta 1 0* has said number of pairs of successive first curves, in which method said fourth expression is set up to express the combined angular extent of said second pitchline curves in terms of theta 1 to total 180/n degrees.
 11. Method of claim 1 for obtaining the profile geometries of a pair of first and second gears of which one symmetrical half of the pitchline of said first gear extends from theta 1 0* to theta 1 180* at the gear axis of symmetry on opposite sides, respectively, of the gear center, and said one symmetrical pitchline half of said first gear has a whole number n of successive lengths of identical curvature and angular extent, of which the one pitchline length starting at theta 1 0* has said number of pairs of successive first curves, and said first and second gears have a gear ratio of n1:n2 which is other than 1:1, in which method said fourth expression is set up to express the combined angular extent of said second pitchline curves in terms of theta 1 to total 360/n2 degrees.
 12. Method of obtaining the profile geometries of a pair of driving and driven balance gears for a pair of driving and driven utility gears, of which said driving gears are adapted to turn in unison, the gears of each pair are meshable at the same given center-to-center distance, and each gear has an axis of symmetry, With one symmetrical half of the pitchline of said driving utility gear being formed by a number of pairs of successive non-circular first curves of known polar equations, of which the curves of each pair have different polar equations and a junction point at a given angle theta 1 from the gear axis of symmetry at which theta 1 is zero, and each of said first curves extends between given angles theta 1, a selected first pitchline curve and its coordinate second pitchline curve of said utility gear having a first given velocity ratio and coordinate pitchline radii r1 and r2, respectively, whose length relationship r1/r2 for any angle theta 1 of r1 has a determinable numerical value, one symmetrical half of the pitchline of said driving balance gear being formed by a number of pairs of successive third curves equal to said number of pairs of successive first pitchline curves, of which each third curve is associated with one of said first pitchline curves and extends between angles theta 3 from the axis of symmetry of said driving balance gear at which theta 3 is zero, with the angles theta 3 between which each third pitchline curve extends being equal to the angles theta 1 between which its associated first pitchline curve extends, a selected one of said third pitchline curves which is associated with said selected first pitchline curve, and its coordinate fourth pitchline curve of said driven balance gear having a second given velocity ratio and radii r3 and r4, respectively, and said selected third pitchline curve and its coordinate fourth pitchline curve satisfy the relation (r1/r2)2 + x2 (r3/r4)2 Z where r1, r2 and r3, r4 are coordinate radii at the same angles theta 1 and theta 3 of r1 and r3, and x and Z are constants of unknown values, and the remaining third pitchline curves have the same polar equations as their associated first pitchline curves, said method comprising, setting up first pairs of equations, of which the equations of each pair express the lengths of the radii of one of said third pitchline curves at angles theta 3 and the lengths of their coordinate pitchline radii of said driven balance gear, and include two different constants of unknown values to arrive at equal pitchline perimeters of said balance gears and at said second given velocity ratio, with the constants in said equation pairs being also different from each other, and the equations of one of said first pairs which express the lengths of the radii of said selected third pitchline curve and of its coordinate fourth pitchline curve have x and Z for their constants and are derived from said relation (r1/r2)2 + x2 (r3/r4)2 Z; setting up a pair of second equations expressing the angular velocities of successive fourth pitchline curves of each pair of said driven balance gear at their function point, with said second equations including the constants in the length expressions of the radii of the respective fourth pitchline curves; setting up a pair of third equations expressing the angles of obliquity of successive third pitchline curves of each pair of said driving balance gear at their junction point, with said third equations including the constants in the length expressions of the radii of the respective third pitchline curves; setting up a fourth equation expressing the combined angular extent of said fourth pitchline curves of said driven balance gear in terms of theta 3, to total 180*, with said fourth equation including the constants in the length expressions of the radii of the respective fourth pitchline curves; expressing in a fifth equation said second given velocity Ratio by the radius relationship of said given third pitchline curve and its coordinate fourth pitchline curve, with said fifth equation including the constants in the length expressions of the radii the respective third and fourth pitchline curves; from said second to fifth equations, and by setting said second and third equations of each pair equal to each other, obtaining the numerical values of all of said constants; and by using the numerical values of said constants in said first pairs of equations, obtaining the actual lengths of coordinate pitchline radii of said balance gears for different angles theta
 3. 13. The method of claim 12, in which said number of pairs of successive first pitchline curves is greater than
 1. 14. The method of claim 13, in which said first pitchline curves are of open and closed types.
 15. The method of claim 12, in which said second given velocity ratio is different from said first given velocity ratio.
 16. The method of claim 12, in which said number of pairs of successive first pitchline curves is greater than 1, said first pitchline curves consist of two end curves and an intermediate curve which is said selected first pitchline curve, and said end curves and their coordinated third pitchline curves of said driving balance gear are selected curves of the same known polar equation.
 17. A pair of meshing gears of one-to-one gear ratio, of which each gear has an axis of symmetry, and each symmetrical half of the pitchline of one of said gears is formed by a number of pairs of successive non-circular first curves of known polar equations, of which the curves of each pair have different polar equations and a junction point whose pitchline radius forms a given angle with the gear axis of symmetry.
 18. A pair of meshing gears as in claim 17, in which said number of pairs of successive curves is
 1. 19. A pair of meshing gears as in claim 17, in which said number of pairs of successive curves is greater than
 1. 20. A pair of meshing gears as in claim 19, in which said curves are of open and closed types.
 21. A pair of meshing gears as in claim 17, in which said first curves consist of two closed end curves and one intermediate open curve, and each symmetrical half of the pitchline of the other gear consists of second curves coordinate with said first curves, with said open first curve and its coordinate second curve having a given velocity ratio.
 22. A pair of meshing gears as in claim 21, in which said open first curve is part of a spiral.
 23. A pair of meshing gears of one-to-one gear ratio, of which the gears have the same plurality of axes of symmetry, with the symmetry axes of each gear passing through the gear center and being equi-angularly spaced, and successive pitchline lengths between the symmetry axes of each gear being of identical curvature, with one of said pitchline lengths of one of said gears being formed by a number of pairs of successive non-circular curves of known polar equations, of which the curves of each pair have different polar equations and a junction point whose pitchline radius forms a given angle with one of the two symmetry axes between which said one pitchline length extends.
 24. A pair of meshing gears of which one gear has a plurality of axes of symmetry passing through the gear center and being equi-angularly spaced, and successive lengths between the symmetry axes of said one gear being of identical curvature, with one of said pitchline lengths being formed by a number of pairs of successive non-circular curves of known polar equations, of which the curves of each pair have different polar equations and a junction point whose pitchline radius forms a given angle with one of the two symmetry axes between which said one pitchline length extends, and the other gear is coordinate with said one gear for a gear ratio between the gears of other than one-to-one.
 25. A pair of meshing balance gears for a pair of meshing utility gears of one-to-one gear ratio and a given center-to-center distance, of which each utility gear has an axis of symmetry, and one symmetrical half of the pitchline of one of said utility gears is formed by a number of pairs of successive non-circular first curves of known polar equations, of which the curves of each pair have different polar equations, each of said first curves extends between given angles theta 1 from the gear axis of symmetry at which theta 1 is zero, and the other utility gear has second pitchline curves coordinate with said first pitchline curves, said balance gears having a one-to-one gear ratio and said given center-to-center distance, and one of said balance gears being adapted for rotation in unison with said one utility gear, with one symmetrical half of the pitchline of said one balance gear being formed by a number of pairs of successive non-circular third curves equal to said number of pairs of successive first pitchline curves, of which each third curve is associated with one of said first pitchline curves and extends between angles theta 3 from the axis of symmetry of said one balance gear at which theta 3 is zero, with the angles theta 3 between which each third pitchline curve extends being equal to the angles theta 1 between which its associated first pitchline curve extends, and the other balance gear has fourth pitchline curves coordinate with said second pitchline curves, one of said first pitchline curves and its coordinate second pitchline curve having radii r1 and r2, respectively, and the one third pitchline curve associated with said one first pitchline curve and its coordinate fourth pitchline curve having radii r 3 and r4, respectively, with said one third pitchline curve and its coordinate fourth pitchline curve satisfying the relationship (r1/r2)2 + x2 (r3/r4)2 Z, where r1, r2 and r3, r4 are coordinate radii at the same angles theta , and theta 3 of r1 and r3, and x and Z are constants of known values, and the remaining third pitchline curves have the same polar equations as their associated first pitchline curves.
 26. A pair of meshing balance gears as in claim 25, in which said number of pairs of successive first pitchline curves is greater than
 1. 27. A pair of meshing balance gears as in claim 25, in which said first pitchline curves consist of two end curves and one intermediate curve, and the third pitchline curve which is associated with said intermediate curve is said one third pitchline curve.
 28. A pair of meshing balance gears as in claim 27, in which said intermediate pitchline curve is an open curve.
 29. A pair of meshing balance gears as in claim 28, in which said open curve is part of a spiral.
 30. A pair of meshing balance gears as in claim 25, in which said one first pitchline curve and its coordinate second pitchline curve have a first given velocity ratio, and said one third pitchline curve and its coordinate fourth pitchline curve have a second given velocity ratio different from said first velocity ratio. 